1. Field of the Invention
The invention relates to music creation, performance and reproduction.
2. Description of the Prior Art
It is common practice that the largest proportion of music is performed on instruments whose notes are tuned according to "EQUAL TEMPERAMENT". By "EQUAL TEMPERAMENT", it means that notes are derived from each octave being logarithmically divided into twelve divisions. This octave, with its twelve notes so divided, is duplicated above and beneath by each corresponding note being multiplied by factors of n.sup.th power of two. Since note "A" is traditionally assigned a frequency of 440 Hz, its octaves above and below by n.sup.th power of two is therefore . . . 110 Hz, 220 Hz, 880 Hz, 1760 Hz . . . etc, where n= . . . -2, -1, 1, 2, . . . The other eleven notes are logarithmically determined to have frequencies: EQU 440*EXP(LOG(2)*n/12)Hz
where for note "A.music-sharp.", n=1; for note "B", n=2; for note "C", n=3; for note "C.music-sharp.", n=4 . . . etc.
The frequencies being calculated of the notes A.music-sharp., B, C, C.music-sharp., D, D.music-sharp., E, F, F.music-sharp., G, G.music-sharp., A' are hence 466.16 Hz, 493.88 Hz, 523.25 Hz, 554.37 Hz, 587.33 Hz, 622.25 Hz, 659.26 Hz, 698.16 Hz, 739.99 Hz, 783.99 Hz, 830.61 Hz and 880.00 Hz. These frequencies extend to the upper and lower octaves with frequencies multiplied by respective multiples of two for an "equal-tempered" keyboard (see FIG. 1).
Historically, there were proposals of other ways of deriving the frequencies of notes. Some of these ways involve arrangements of the scale such that the frequencies of its notes form simple ratios with one another, and are generally termed "PURE INTONATION". However, this has long been found to be theoretically and practically impossible to apply to instruments of pre-determined tuning, since it will result in "pure" intervals only between particular pairs of notes, but involves heavy penalties for other intervals. It has also been long realized that such formation will result in music performances that are rather unpleasant to the ears of a listener.
Furthermore, there exists no reliable theory to explain how music becomes "pleasant" to listen to. The sensors of the ear remain one of the most obscure subjects in the area of scientific research.
On the other hand, it is known that good violinists and singers do adjust each note's frequency during performances in real time so that each note deviates a little away from the "equally-tempered" frequencies, and hence render their performances significantly more pleasant to the listener. The converse is also true that poor performers appear to adjust in wrong ways and render their performances musically unpleasant. Very many proposals have been made over hundreds of years as to how notes scales and instruments should be tuned or adjusted. However, none of these proposals has yet been found compatible to aesthetically pleasing performances in the realms of classical, popular or traditional music.
For example, the following is such a proposal of a standard scale that fails to fulfil a requirement for good listening:
do 9/8 re 10/9 me 16/15 fa 9/8 so 10/9 la 9/8 te 16/15 do PA1 do 9/8 re 10/9 me 16/15 fa 9/8 so 10/9 la 9/8 te 16/15 do, the scale now becomes: PA1 do 10/9 re.sub.1 81/80 re.sub.2 10/9 me . . . etc
This scale generates "good" major thirds (9/8*10/9=5/4), between note-pairs do/me, fa/la, so/te, and "good" minor thirds (9/8*16/15=6/5) between te/re, me/so, la/do. However, the scale produces a "bad" minor third (10/9*16/15=32/27) between re/fa. The scale also produces "Good" perfect fifth (9/8*10/9*9/8*16/15=3/2) between do/so and me/te, but "bad" fifth (10/9*16/15*9/8*10/9=40/27) between re/la. Performances in this scale arrangement sound strange and unnatural.
Historically, therefore, attempts were made to solve the problem of intonation on fixed-pitch instruments, in particular, keyboard instruments, by using multiple keys for each of the twelve notes in each octave. For example, the "bad" minor thirds (10/9*16/15=32/27) between re/fa may be overcome by the insertion of an extra key for re, so that instead of the:
Additionally, the "bad" minor thirds (10/9*16/15=32/27) between re/fa is substituted by re.sub.1 /fa (10/9*81/80*16/15=6/5) instead, giving a "good" minor third.
Although such solutions may solve apparent "problems" for isolated instances of the sounding of two notes, it does not address the needs of harmonic music flow, nor offer any explanation why listeners, including children, can readily point out wrong notes in very complicated polyphony.
Instead of splitting up keys with its complication of more than the original twelve keys for each octave, another solution is providing more than one stave of keys for the instrument. Instruments such as the harpsichord and the pipe organ do in fact have more than one stave of keys. However, there is not found any record of any workable proposal of how these staves should be tuned so that the problem of intonation may be overcome.